# Definition:Jordan Arc

## Definition

Let $f: \closedint 0 1 \to \R^2$ be an injective path in the Euclidean plane.

Then $f$ is called a **Jordan arc**.

## Also known as

Some texts refer to a **Jordan arc** as merely an **arc**.

## Also defined as

Some texts define a **Jordan arc** $f: \closedint 0 1 \to X$ as an injective path, where $X$ is alternatively defined as:

- the complex plane $\C$
- a real Euclidean space $\R^n$
- a $T_2$ (Hausdorff) topological space $\struct{ S, \tau_S }$

This is what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an arc.

Some texts, especially those on complex analysis, drop the condition about injectivity and instead state that:

- $\map f {t_1} \ne \map f {t_2}$ for all $t_1 ,t_2 \in \hointr 0 1$ with $t_1 \ne t_2$

- $\map f t \ne \map f 1$ for all $t \in \openint 0 1$

That is, either $f$ is injective and a **Jordan Arc**, or $\map f 0 = \map f 1$, when $f$ is a Jordan curve by the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.

Some texts, especially those on topology, define a **Jordan arc** as topological subspace $\struct{C, \tau_C}$ of $\R^2$ or $X$, where $\struct{C, \tau_C}$ is homeomorphic to the closed interval $\closedint 0 1$.

This means they consider a **Jordan arc** to be a topological space rather than a mapping.

## Also see

## Source of Name

This entry was named for Marie Ennemond Camille Jordan.

## Sources

- 2017: Thierry Vialar:
*Handbook of Mathematics*: $10$: Topology: $\S 14$: Theory of Curves